A pendulum has maximum kinetic energy `K_1` If its length is doubled keeping amplitude same then maximum kinetic energy becomes `K_2` .Then relation between `K_1 and K_2` is

Options

A. `K_2=2K_1`
B. `K_1=2K_2`
C. `K_2=K_1`
D. `K_1=4K_2`

Answer

To solve the problem, we need to analyze the relationship between the maximum kinetic energy of a pendulum and its length. Here's a step-by-step solution:

### Step 1: Understand the Concept of Maximum Kinetic Energy
In a simple harmonic motion, the maximum kinetic energy (K) of a pendulum occurs at the mean position (the lowest point of the swing). At this point, all the potential energy has been converted into kinetic energy.

**Hint:** Remember that kinetic energy is maximum at the mean position and potential energy is maximum at the extreme positions.

### Step 2: Write the Expression for Maximum Kinetic Energy
The maximum kinetic energy (K) of a pendulum can be expressed as:
\[ K = \frac{1}{2} mv^2 \]
where \( m \) is the mass of the pendulum bob and \( v \) is the maximum velocity at the mean position.

**Hint:** Kinetic energy depends on the mass and the square of the velocity.

### Step 3: Relate Velocity to Length and Amplitude
The maximum velocity \( v \) of the pendulum bob can be derived from the conservation of energy. The potential energy at the maximum height (when the pendulum is at amplitude \( A \)) is converted to kinetic energy at the mean position. The height \( h \) can be expressed in terms of the length \( L \) and the angle \( \theta \):
\[ h = L - L \cos(\theta) = L(1 - \cos(\theta)) \]
Thus, the potential energy at the maximum height is:
\[ PE = mgh = mgL(1 - \cos(\theta)) \]

At the mean position, this potential energy converts to kinetic energy:
\[ K = mgL(1 - \cos(\theta)) \]

**Hint:** Use the relationship between height and length to express potential energy.

### Step 4: Analyze the Effect of Doubling the Length
If the length of the pendulum is doubled (let's denote the new length as \( L_2 = 2L \)), while keeping the amplitude \( A \) the same, the new maximum kinetic energy \( K_2 \) can be calculated as:
\[ K_2 = mg(2L)(1 - \cos(\theta)) = 2mgL(1 - \cos(\theta)) = 2K_1 \]

**Hint:** Notice how the kinetic energy scales with the length of the pendulum.

### Step 5: Establish the Relationship Between \( K_1 \) and \( K_2 \)
From the above analysis, we can conclude that:
\[ K_2 = 2K_1 \]

**Hint:** This shows that the maximum kinetic energy is directly proportional to the length of the pendulum.

### Final Answer
The relationship between the maximum kinetic energy before and after doubling the length of the pendulum is:
\[ K_2 = 2K_1 \]